The goal of this paper is to show that a wide class of Harish-Chandra ( g , K ) (\mathfrak {g},K) -modules including all irreducible ones come with a certain canonical filtration.

We consider the symplectic group S p 2 n Sp_{2n} defined over a p p -adic field F F , where p = 2 p=2 . We prove that every simple supercuspidal representation (in the sense of Gross–Reeder) of S p 2 n ( F ) Sp_{2n}(F) corresponds to an irreducible L L -parameter under the local Langlands correspondence for S p 2 n Sp_{2n} established by Arthur.

In this paper, we explore natural connections among the representations of the extended affine Lie algebra s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) with C q = C q [ t 0 ± 1 , t 1 ± 1 ] \mathbb {C}_q=\mathbb {C}_q[t_0^{\pm 1},t_1^{\pm 1}] an irrational quantum 2 2 -torus, the simple affine vertex algebra L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) with ℓ \ell a positive integer, and Levi subgroups G L I \mathrm {GL}_{\mathbf {I}} of G L ℓ ( C ) \mathrm {GL}_\ell (\mathbb {C}) . First, we give a canonical isomorphism between the category of integrable restricted s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) -modules of level ℓ \ell and that of equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules. Second, we classify irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules. Third, we establish a duality between irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -modules and irreducible regular G L I \mathrm {GL}_{\mathbf {I}} -modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible N \mathbb N -graded equivariant quasi L s l ∞ ^ ( ℓ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0) -module. Fifth, we completely determine the following branchings: (i) The branching from L s l ∞ ^ ( ℓ , 0 ) ⊗ L s l ∞ ^ ( ℓ ′ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell ,0)\otimes L_{\widehat {\mathfrak {sl}_\infty }}(\ell ’,0) to L s l ∞ ^ ( ℓ + ℓ ′ , 0 ) L_{\widehat {\mathfrak {sl}_\infty }}(\ell +\ell ’,0) for quasi modules. (ii) The branching from s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) to its Levi subalgebras. (iii) The branching from s l N ^ ( C q ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q) to its subalgebras s l N ^ ( C q [ t 0 ± M 0 , t 1 ± M 1 ] ) \widehat {\mathfrak {sl}_N}(\mathbb {C}_q[t_0^{\pm M_0},t_1^{\pm M_1}]) .

We study the mod ℓ \ell Weil representation of a finite unitary group and related Deligne–Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod ℓ \ell Howe correspondence for ( S p 2 n , O 2 − ) (\mathrm {Sp}_{2n},\mathrm {O}_2^-) including the case where p = 2 p=2 .

We define and study an analogue of Category O \mathcal {O} in the context of Kazhdan and Laumon’s gluing construction for perverse sheaves on the basic affine space. We explicitly describe the simple objects in this category, and we show its linearized Grothendieck group is isomorphic to a natural submodule of Lusztig’s periodic Hecke module. We then provide a categorification of these results by showing that the Kazhdan-Laumon Category O \mathcal {O} is equivalent to a full subcategory of a suitably-defined category of perverse sheaves on the semi-infinite flag variety.